Subgroups — C₂×#21426371 #28744166

Order 15104.

558 conjugacy classes of subgroups.

The subgroup lattice has 152 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16837807	C₂	1	2	yes	maximal
2	#21426371	C₂	1	8	yes	maximal, direct product factor
3	C₂×#16837228	C₂	1	4	yes	maximal
4	C₂×C₄×C₁₆×C₅₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#14273727	—	2	2	no	—
6	#16837807	—	2	4	no	—
7	#16837807	C₂²	1	8	yes	—
8	C₂×#14273661	—	2	2	no	—
9	C₂×#14273661	C₂²	1	4	yes	—
10	#16837228	C₂²	1	16	yes	—
11	C₂²×C₁₆×C₅₉	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
13	C₄×C₁₆×C₅₉	C₂²	1	4	yes	—
14	C₂²×Dic₁₁₈	—	4	2	no	—
15	#14273727	—	2	8	no	—
16	#14273727	—	4	4	no	—
17	C₂×C₄⋊Dic₅₉	—	2	4	no	—
18	C₂×C₄⋊Dic₅₉	—	4	2	no	—
19	#14273661	—	2	8	no	—
20	#14273661	—	2	4	no	—
21	#14273661	C₂³	1	8	yes	—
22	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
23	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
24	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
25	C₂×C₁₆×C₅₉	Q₈	1	8	yes	complemented
26	C₂×C₁₆×C₅₉	C₂³	1	4	yes	—
27	C₄×C₈×C₅₉	D₄	1	2	yes	—
28	C₄×C₈×C₅₉	C₂³	1	2	yes	—
29	C₂²×C₄×C₅₉	D₈	1	1	yes	characteristic
30	C₂²×C₄×C₅₉	D₈	1	1	yes	characteristic
31	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
32	C₂²×Dic₅₉	—	4	4	no	—
33	C₂×Dic₁₁₈	—	4	8	no	—
34	C₂×Dic₁₁₈	—	8	8	no	—
35	C₂×C₈×C₅₉	Q₁₆	1	4	yes	—
36	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
37	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
38	C₂×C₈×C₅₉	C₂×Q₈	1	4	yes	—
39	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
40	C₄²×C₅₉	D₈	1	2	yes	—
41	C₄²×C₅₉	C₂×D₄	1	2	yes	—
42	C₄⋊Dic₅₉	—	2	8	no	—
43	C₄⋊Dic₅₉	—	4	4	no	—
44	C₄⋊Dic₅₉	—	4	8	no	—
45	C₁₆×C₅₉	C₂×Q₈	1	8	yes	complemented
46	C₂³×C₅₉	C₂×D₈	1	1	yes	characteristic
47	C₂×C₄×C₅₉	D₁₆	1	4	yes	—
48	C₂×C₄×C₅₉	Q₃₂	1	4	yes	—
49	C₂×C₄×C₅₉	C₄⋊Q₈	1	2	yes	—
50	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
51	C₂×C₄×C₅₉	C₂×D₈	1	2	yes	—
52	C₂×C₄×C₅₉	C₂×D₈	1	2	yes	—
53	C₂×C₄×C₅₉	C₂×Q₁₆	1	2	yes	—
54	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
55	C₂×Dic₅₉	—	4	8	no	—
56	C₂×Dic₅₉	—	8	8	no	—
57	Dic₁₁₈	—	8	16	no	—
58	C₈×C₅₉	C₄⋊Q₈	1	2	yes	—
59	C₈×C₅₉	C₂×Q₁₆	1	4	yes	—
60	C₈×C₅₉	C₂²×Q₈	1	2	yes	—
61	C₂×#2252	—	59	1	no	maximal, Sylow
62	C₂²×C₅₉	#500	1	2	yes	—
63	C₂²×C₅₉	C₂×D₁₆	1	2	yes	—
64	C₂²×C₅₉	C₂×Q₃₂	1	2	yes	—
65	C₂²×C₅₉	C₂²×D₈	1	1	yes	characteristic
66	C₄×C₅₉	#500	1	2	yes	—
67	C₄×C₅₉	C₂×D₁₆	1	4	yes	—
68	C₄×C₅₉	C₂×Q₃₂	1	4	yes	—
69	C₄×C₅₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
70	C₄×C₅₉	C₂²×Q₁₆	1	1	yes	characteristic
71	Dic₅₉	—	8	8	no	—
72	C₂×#500	—	59	2	no	—
73	#2252	—	59	8	no	—
74	C₂×#366	—	59	4	no	—
75	C₂×C₄×C₁₆	D₅₉	1	1	yes	characteristic
76	C₂×C₅₉	#2252	1	4	yes	complemented
77	C₂×C₅₉	C₂×#500	1	1	yes	characteristic
78	C₂×C₅₉	C₂²×D₁₆	1	1	yes	characteristic
79	C₂×C₅₉	C₂²×Q₃₂	1	1	yes	characteristic
80	C₂×C₄⋊Q₈	—	118	2	no	—
81	C₂²×C₁₆	C₂×D₅₉	1	2	yes	—
82	#500	—	59	8	no	—
83	#500	—	118	4	no	—
84	C₂×#107	—	59	4	no	—
85	C₂×#107	—	118	2	no	—
86	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
87	#366	—	59	16	no	—
88	C₄×C₁₆	C₂×D₅₉	1	4	yes	—
89	C₅₉	C₂×#2252	1	1	yes	characteristic, complemented, Sylow
90	C₂²×Q₈	—	236	2	no	—
91	C₂²×C₈	D₂₃₆	1	1	yes	characteristic
92	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
93	C₄⋊Q₈	—	118	8	no	—
94	C₄⋊Q₈	—	236	4	no	—
95	C₂×C₄⋊C₄	—	118	4	no	—
96	C₂×C₄⋊C₄	—	236	2	no	—
97	C₂×C₄²	D₂₃₆	1	1	yes	characteristic
98	C₂×C₁₆	Dic₁₁₈	1	8	yes	complemented
99	C₂×C₁₆	C₂²×D₅₉	1	4	yes	—
100	#107	—	59	8	no	—
101	#107	—	118	4	no	—
102	#107	—	118	8	no	—
103	C₄×C₈	D₂₃₆	1	2	yes	—
104	C₄×C₈	C₂²×D₅₉	1	2	yes	—
105	C₂×Q₈	—	472	8	no	—
106	C₂×Q₈	—	236	8	no	—
107	C₂²×C₄	—	236	4	no	—
108	C₂²×C₄	D₄₇₂	1	1	yes	characteristic
109	C₂²×C₄	D₄₇₂	1	1	yes	characteristic
110	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
111	C₂×C₈	Dic₂₃₆	1	4	yes	—
112	C₂×C₈	C₂×Dic₁₁₈	1	4	yes	—
113	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
114	C₂×C₈	C₂×D₂₃₆	1	2	yes	—
115	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
116	C₄⋊C₄	—	118	8	no	—
117	C₄⋊C₄	—	236	4	no	—
118	C₄⋊C₄	—	236	8	no	—
119	C₄²	D₄₇₂	1	2	yes	—
120	C₄²	C₂×D₂₃₆	1	2	yes	—
121	C₁₆	C₂×Dic₁₁₈	1	8	yes	complemented
122	C₂³	C₂×D₄₇₂	1	1	yes	characteristic
123	Q₈	—	472	16	no	—
124	C₂×C₄	—	472	8	no	—
125	C₂×C₄	—	236	8	no	—
126	C₂×C₄	D₉₄₄	1	4	yes	—
127	C₂×C₄	Dic₄₇₂	1	4	yes	—
128	C₂×C₄	#14273727	1	2	yes	—
129	C₂×C₄	C₂×D₄₇₂	1	1	yes	characteristic
130	C₂×C₄	C₂×D₄₇₂	1	2	yes	—
131	C₂×C₄	C₂×D₄₇₂	1	2	yes	—
132	C₂×C₄	C₂×Dic₂₃₆	1	2	yes	—
133	C₂×C₄	C₂²×D₂₃₆	1	1	yes	characteristic
134	C₈	#14273727	1	2	yes	—
135	C₈	C₂×Dic₂₃₆	1	4	yes	—
136	C₈	C₂²×Dic₁₁₈	1	2	yes	—
137	C₂²	#16837807	1	2	yes	—
138	C₂²	C₂×D₉₄₄	1	2	yes	—
139	C₂²	C₂×Dic₄₇₂	1	2	yes	—
140	C₂²	C₂²×D₄₇₂	1	1	yes	characteristic
141	C₄	—	472	8	no	—
142	C₄	#16837807	1	2	yes	—
143	C₄	C₂×D₉₄₄	1	4	yes	—
144	C₄	C₂×Dic₄₇₂	1	4	yes	—
145	C₄	C₂×#14273727	1	1	yes	characteristic
146	C₄	C₂²×Dic₂₃₆	1	1	yes	characteristic
147	C₂	#21426371	1	4	yes	direct product factor
148	C₂	C₂×#16837807	1	1	yes	characteristic
149	C₂	C₂²×D₉₄₄	1	1	yes	characteristic
150	C₂	C₂²×Dic₄₇₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₄₇₂
commutator, cosocle	C₂×C₈×C₅₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₉
Fitting	C₂×C₄×C₁₆×C₅₉	C₂
socle, prime radical	C₂³×C₅₉	C₂×D₈
hypercenter	C₂×C₄×C₁₆	D₅₉
hypocenter	C₅₉	C₂×#2252
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. #21426371	C₂	direct (N × H)
25. C₂×C₁₆×C₅₉	Q₈	semidirect
45. C₁₆×C₅₉	C₂×Q₈	semidirect
76. C₂×C₅₉	#2252	semidirect
89. C₅₉	C₂×#2252	semidirect
98. C₂×C₁₆	Dic₁₁₈	semidirect
121. C₁₆	C₂×Dic₁₁₈	semidirect
147. C₂	#21426371	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#16837807	2
#21426371	2
C₂×#16837228	2
C₂×C₄×C₁₆×C₅₉	2
C₂×#2252	59

Subgroups — C2×#21426371 #28744166

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21426371 #28744166